Journal of Mathematical Physics, Analysis, Geometry 2020, vol. 16, No 2, pp. 119-137   https://doi.org/10.15407/mag16.02.119     ( to contents , go back )

### On Certain Geometric Properties in Banach Spaces of Vector-Valued Functions

Jan-David Hardtke

University of Leipzig, 10 Augustusplatz, Leipzig, 04109, Germany
E-mail: hardtke@math.uni-leipzig.de

Received May 9, 2019, revised June 6, 2019.

Abstract

We consider a certain type of geometric properties of Banach spaces, which includes, for instance, octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem, which, roughly speaking, states the following: if the property in question is stable under certain finite absolute sums (for example, finite $\ell^p$-sums), then it is also stable under the formation of corresponding Köthe-Bochner spaces (for example, $L^p$-Bochner spaces). From this general theorem, we obtain as corollaries a number of new results as well as some alternative proofs of already known results concerning octahedral and almost square spaces and their relatives, diameter two properties, lush spaces and other classes.

Mathematics Subject Classification 2000: 46B20, 46E40
Key words: absolute sums, Köthe-Bochner spaces, Lebesgue-Bochner spaces, octahedral spaces, almost square spaces, diameter two properties, lush spaces, generalised lush spaces, Daugavet property